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Study Materials › Arithmetic › Percentages
Percentage is the single most useful topic in the arithmetic section — profit & loss, interest, discount and data interpretation are all percentage problems wearing different clothes. Get the fraction equivalents into memory and a large part of the paper becomes mental arithmetic.
x% of N = N × x / 100
What percent is a of b = (a / b) × 100
Percentage change = (New − Old) / Old × 100
Increase by x% → multiply by (100 + x)/100
Decrease by x% → multiply by (100 − x)/100
Learn this table by heart. Converting 37½% into 3/8 turns a two-minute calculation into a five-second one.
| Fraction | Percentage | Fraction | Percentage |
|---|---|---|---|
| 1/2 | 50% | 1/9 | 11⅙% |
| 1/3 | 33⅓% | 1/10 | 10% |
| 1/4 | 25% | 1/11 | 9 1/11% |
| 1/5 | 20% | 1/12 | 8⅓% |
| 1/6 | 16⅔% | 1/16 | 6¼% |
| 1/7 | 14 2/7% | 1/20 | 5% |
| 1/8 | 12½% | 1/25 | 4% |
| 3/8 | 37½% | 5/8 | 62½% |
| 2/3 | 66⅔% | 3/4 | 75% |
Two successive changes of a% and b% combine into a single change of:
a + b + ab/100 (use a negative sign for a decrease)
A 20% rise followed by a 20% fall is not zero — it is 20 − 20 − 400/100 = −4%, a net loss. This trap appears in almost every paper.
If 22½% increase of a number is 98, then 22½% decrease of that number is
Let the number be N. An increase of 22½% means N × 1.225 = 98, so N = 98 / 1.225 = 80.
A decrease of 22½% gives 80 × 0.775 = 62.
Faster: the number itself is never needed twice — note that
(100 − 22.5) / (100 + 22.5) = 77.5 / 122.5, so the answer is 98 × 77.5 / 122.5 = 62 directly.
The marked price of an article shows an increase of 15% on its cost price. Later this marked price was decreased by 20%. If the cost price is ₹1,200, find the profit or loss.
Marked price = 1200 × 1.15 = 1380. Selling price = 1380 × 0.80 = 1104.
Loss = 1200 − 1104 = 96, so loss% = 96 / 1200 × 100 = 8% loss.
Successive-change method: +15% then −20% gives
15 − 20 − (15 × 20)/100 = −8%. Same answer, no multiplication.
If 123 more articles can be bought for a certain amount when the price of an article is decreased by 33⅓%, how many articles could have been bought without the discount?
33⅓% = 1/3, so the new price is 2/3 of the old. Price and quantity are inversely proportional, so the
quantity becomes 3/2 of the original — an increase of 1/2.
That extra half equals 123 articles, so the original number = 246.
Next: Profit & Loss · Simple & Compound Interest · Time & Work